Question

Evaluate the sums in Problems 5-12 using the formula for the sum of an arithmetic series. $$ \sum_{i=1}^{30}(2 i+10) $$

Short answer

Question: Calculate the sum of the arithmetic series given by the sequence (2i + 10), where i goes from 1 to 30. Answer: The sum of the given arithmetic series is 1230.

Step-by-step solution

Identify the arithmetic sequence

The given arithmetic sequence is \((2i + 10)\), where \(i\) goes from \(1\) to \(30\).

Determine the number of terms in the series

Since \(i\) goes from \(1\) to \(30\), there are a total of \(30\) terms in this series.

Calculate the first and last term of the series

Now, we will calculate the first and last term of the series to use the formula for the sum: - First term (\(a_1\)): Substitute \(i=1\) into the sequence formula: \(a_1 = 2(1) + 10 = 12\) - Last term (\(a_{30}\)): Substitute \(i=30\) into the sequence formula: \(a_{30}=2(30) + 10 = 70\)

Use the formula for the sum of an arithmetic series

The formula for the sum of an arithmetic series is given by: $$ S_n = \frac{n(a_1 + a_n)}{2} $$ where \(S_n\) is the sum of the series, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term. Now, we will plug in values we've obtained into the formula: $$ S_{30} = \frac{30(12 + 70)}{2} $$

Calculate the sum of the series

Solve the equation to find the sum of the series: $$ S_{30} = \frac{30(82)}{2} = 15(82) = 1230 $$ So, the sum of the series is \(1230\).

Key concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence is a concept used to find the total of all numbers in a specific arithmetic series. An arithmetic sequence is a series of numbers in which the difference between any two successive terms is constant. For example, in the sequence \(2, 4, 6, 8, \ldots\), the difference between each term is \(2\).
To calculate the sum of an arithmetic sequence, we use the formula:

• \( S_n = \frac{n(a_1 + a_n)}{2} \)

The formula effectively averages the first and last terms of the sequence and then multiplies by the number of terms \(n\) to get the total sum. This approach works because the sequence forms a predictable pattern.
For the exercise, the arithmetic sequence is \((2i + 10)\), with \(i\) ranging from \(1\) to \(30\). By substituting into the sum formula, we determined that the sum \(S_{30}\) is \(1230\). This means that adding all terms in the sequence results in the value \(1230\).

Number of Terms in Sequence

Understanding how to determine the number of terms in an arithmetic sequence is crucial. The number of terms, denoted as \(n\), refers to how many numbers there are in the series. It's especially important for calculating the sum of the sequence.
In the exercise, the sequence term formula is \((2i + 10)\) with \(i\) starting at \(1\) and ending at \(30\). Clearly, the sequence has \(30\) terms because we count each integer value of \(i\) from \(1\) to \(30\).
This information is essential when using the sum formula \(S_n = \frac{n(a_1 + a_n)}{2}\), since \(n\) tells us the number of times we should multiply our average of the first and last terms. Without knowing \(n\), we cannot accurately compute the sequence's sum.

First and Last Terms

Identifying the first and last terms of an arithmetic sequence is an important step to calculate its sum. The first term, \(a_1\), is found by substituting the smallest value of \(i\) into the sequence formula. Similarly, the last term, \(a_n\), is found by using the largest value of \(i\).
For our exercise, to find the first term \(a_1\), substitute \(i = 1\) into the sequence formula \((2i + 10)\), resulting in \(a_1 = 12\).
To find the last term \(a_{30}\), substitute \(i = 30\) into the same formula, resulting in \(a_{30} = 70\).
With these terms identified, we can use them in the sum formula \(S_n = \frac{n(a_1 + a_n)}{2}\). Here, \(a_1 = 12\) and \(a_{30} = 70\) are directly plugged into the formula, helping us conclude the sum of the series is \(1230\). Knowing the first and last terms highlights the symmetry in arithmetic sequences, as they help us work towards finding the total sum efficiently.